Consider the following weighted undirected graph. 9 A 8 10 3 D E 14 F 2, 16 15 G 12 13 (a) the cheapest edge crossing the cut. Explain why edge (D, H) is safe. In other words, give a cut where the edge is (b) appearing in the Minimum Spanning Tree (MST) in the order they are added to the MŞT. We would like to run the Kruskal's algorithm on this graph. List the edges

Consider the following weighted undirected graph. 9 A 8 10 3 D E 14 F 2, 16 15 G 12 13 (a) the cheapest edge crossing the cut. Explain why edge (D, H) is safe. In other words, give a cut where the edge is (b) appearing in the Minimum Spanning Tree (MST) in the order they are added to the MŞT. We would like to run the Kruskal's algorithm on this graph. List the edges

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Prove by contradiction or counterexample. Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.
Consider a graph G that is comprised only of non-negative weight edges such that (u, v) € E, w(u, w) > 0. Is it possible for Bellman-Ford and Dijkstra's algorithm to produce different shortest path trees despite always producing the same shortest-path weights? Justify your answer.
. Let G be a weighted, connected, undirected graph, and let V1 and V2 be a partition of the vertices of G into two disjoint nonempty sets. Furthermore, let e be an edge in the minimum spanning tree for G such that e has one endpoint in V1 and the other in V2. Give an example that shows that e is not necessarily the smallest- weight edge that has one endpoint in V1 and the other in V2.
Let G = (V, E) be a connected, undirected graph, and let s be a fixed vertex in G. Let TB be the spanning tree of G found by a bread first search starting from s, and similarly TD the spanning tree found by depth first search, also starting at s. (As in problem 1, these trees are just sets of edges; the order in which they were traversed by the respective algorithms is irrelevant.) Using facts, prove that TB = TD if and only if TB = TD = E, i.e., G is itself a tree.
An optimization version of the Galactic Shortest-Path problem in the homework isdescribed as follows. Let G = (V, E) be a connected undirected graph in which each edge e \in Ehas an integer length l(e) >= 0 and an integer risk r(e) >= 0 Suppose s is the source node andt is the destination node. For any s - t path in G, its total length (resp., risk) is the sum of thelengths (resp., risks) of all its edges. An s-t path is said to be safe if its total risk is at mosta given positive integer R. The goal is to find a shortest safe s-t path if there is any. Give anO (ER)-time dynamic programming algorithm for solving this problem
let us take any standard graph G=(v,e) and let us pretend each edge is the same exact weight. let us think about a minimum spanning tree of the graph G, called T = (V, E' ). under each part a and b illustrate then show that a) s a unique path between u and v in T for all u, v ∈ V . b) tree T is not unique. provide proof
a) Draw the connected subgraph of the given graph above which contains only four nodes ACGB and is also a minimum spanning tree with these four nodes. What is its weighted sum? Draw the adjacency matrix representation of this subgraph (use boolean matrix with only 0 or 1, to show its adjacency in this case).b) Find the shortest path ONLY from source node D to destination node G of the given graph above, using Dijkstra’s algorithm. Show your steps with a table as in our course material, clearly indicating the node being selected for processing in each step.c) Draw ONLY the shortest path obtained above, and indicate the weight in each edge in the diagram. Also determine the weighted sum
Any given connected graph, G, after Traversal, all the paths passed through will get a Spanning tree. It is known that the Traversal of Graph is divided into DFS and BFS. Please discuss whether the Spanning Tree of DFS is unique? Why? In the same way, it is also possible to discuss whether G's BFS Spanning Tree is unique? Why? can you give me some simple opinion?
Let G = (V,E) be a simple graph which is weighted, undirected, and connected. Sup- pose G contains a unique edge having the largest weight. Let emax be this edge. Suppose removing emax in G does not disconnect G. Prove that any minimum spanning tree of G must not contain the edge emax. Hint: Prove by contradiction. (Consider on the contrary that the MST contains emax. Think in this way: Can we replace this edge by some other edge to reduce the total weight?)
True or false: let G be an arbitrary connected, undirected graph with a distinct cost c(e) on every edge e. suppose e* is the cheapest edge in G; that is, c(e*) <c(e) for every edge e is not equal to e*. Any minimum spanning tree T of G contains the edge e*
Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.
Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let P be the shortest path between two nodes s, t. Now, suppose we replace each edge weight ℓ(e) withℓ(e)^2, then P is still a shortest path between s and t.